There is not a lot to find about permits, but you can find where you need to pay for entrance, and photo and video cameras.
In the extended text you can find the text:
Govind Wildlife Sanctuary Information Center
Govind Pashu Vihar Sanctuary,
Divisional Forest Officer,
Tons Forest Division.
In this travel company's track itinerary and facts, they do mention that permits are included into what you pay for.
That confirms that you need to have a permit.
From how hard it is to find any information I would suggest looking into doing the track with a company.
From this page I did not find anything about needing a tour company, but also not about permits.
As I have never been in the area, an internet search is the best I can do for you.
I did find that Sankri did not bring in many results, searches on Har-Ki-Doon and its alternate spelling Har Ki Dun gave more help. But searching on either separately might be best.